Universal coset field theory

We formulate a universal coset field theory, a four-dimensional Euclidean quantum field theory whose exceptional datum is the Albert algebra \(\mathbb{J}\) and the associated compact Lie type \(E_{6}\). Its central result is that, under the two stated axioms, gravity is not a fundamental interaction: spacetime and the metric that propagates on it are emergent, induced structures rather than postulated inputs, and no fundamental graviton field ever appears. An \(E_{6}\)-invariant coercive potential has vacuum orbit Cartan's Hermitian symmetric space EIII,
\[
E_{6}/(\mathrm{Spin}(10)\times U(1)).
\]
Its \(32\) real tangent directions form the irreducible \(\mathrm{SO}(10)\times U(1)\) multiplet with complexification
\[
\mathbf{16}_{-3}\oplus\overline{\mathbf{16}}_{+3}
\]
and carry a unique positive invariant metric. For nondegenerate spacetime-filling maps into EIII, the pull-back of this metric gives a Riemannian first fundamental form, so the spacetime metric is derived from the coset field configuration rather than assumed \textit{a priori}. Lorentzian physics is interpreted through Osterwalder--Schrader reconstruction, and one-loop matter determinants induce an Einstein--Hilbert term in the sense of Sakharov: gravity is thus a radiative effect of matter loops on the coset, not a fundamental field, at every stage of the construction.

We compare this coset sector with the gauge-unfixed Higgs orbit of an \(E_{8}\times E_{8}\) heterotic compactification with visible \(E_{6}\). At the level of the two-derivative bosonic action and the \((\mathrm{SO}(10)\times U(1))_{\psi}\) current algebra, the visible sector matches; a \(\mathbb{Z}_{12}\) asymmetric orbifold with three generations and an adjoint Higgs supplies the candidate realization. The resulting Albert--EIII criterion is relative: within the Albert-selected heterotic class, a theory can realize the correspondence only if its residual Higgs/moduli sector is the EIII coset with the stated tangent representation, metric, gauge data, anomaly and Green--Schwarz matching, and worldsheet consistency. In the stated truncation, the functional renormalization-group flow has a unique ultraviolet-attractive non-Gaussian fixed point and gives
\[
M_{\mathrm{GUT}}\simeq 2\times 10^{16}\,\mathrm{GeV}.
\]
We also show that the Riemann hypothesis is equivalent to the absence of complex mass-squared modes, which is a spectral reformulation of the prime-orbit trace of the finite adelic zeta sector.

We distinguish the representation-theoretic consequences from assumptions requiring further dynamical input.